Standard +0.3 This is a standard FP1 technique for finding complex square roots algebraically by equating real and imaginary parts, then solving simultaneous equations. While it requires careful algebra with surds, it's a routine textbook exercise with a well-established method that students practice regularly. The presence of √5 adds minor computational complexity but doesn't require novel insight.
3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
Attempt to equate real and imaginary parts of \((x+iy)^2\) and \(11 + 12\sqrt{5}\)
A1
Obtain both results cao
M1*
Obtain a quadratic in \(x^2\) or \(y^2\)
\(\pm(2\sqrt{5} + 3i)\)
DM1
Solve a 3 term quadratic to obtain a value for \(x\) or \(y\)
A1
Obtain 1 correct answer as complex number
A1
Obtain only the other correct answer
[6]
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - y^2 = 11$ and $xy = 6\sqrt{5}$ | M1 | Attempt to equate real and imaginary parts of $(x+iy)^2$ and $11 + 12\sqrt{5}$ |
| | A1 | Obtain both results cao |
| | M1* | Obtain a quadratic in $x^2$ or $y^2$ |
| $\pm(2\sqrt{5} + 3i)$ | DM1 | Solve a 3 term quadratic to obtain a value for $x$ or $y$ |
| | A1 | Obtain 1 correct answer as complex number |
| | A1 | Obtain only the other correct answer |
| **[6]** | | |
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3 Use an algebraic method to find the square roots of $11 + ( 12 \sqrt { 5 } ) \mathrm { i }$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers.
\hfill \mbox{\textit{OCR FP1 2013 Q3 [6]}}